# GraphHub

GraphHub[g]

gives the set of vertices with maximum vertex degree in the underlying simple graph of g.

GraphHub[g,"In"]

gives the set of vertices with maximum vertex in-degree.

GraphHub[g,"Out"]

gives the set of vertices with maximum vertex out-degree.

GraphHub[{vw,},]

uses rules vw to specify the graph g.

# Details • The vertex degree for a vertex v is the number of edges incident to v.
• For a directed graph, the in-degree is the number of incoming edges and the out-degree is the number of outgoing edges.
• For an undirected graph, an edge is taken to be both an in-edge and out-edge.

# Examples

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## Basic Examples(1)

Find the hub for a graph:

Highlight:

## Scope(5)

GraphHub works with undirected graphs:

Directed graphs:

Use rules to specify the graph:

Compute the hubs with respect to vertex in-degree and vertex out-degree:

GraphHub works with large graphs:

## Applications(7)

The administrator is the hub of the friendship network between members of a karate club:

Find the people with the most family members present at the family gathering:

Find the publications with the most citations in a citation network:

With the most references to other publications:

The terrorist network linked to the tragic events of September 11, 2001. The ringleader of the conspiracy is the hub of the network:

The Medici family is the hub of the marriage network of the ruling families of Florence:

It is the most powerful family and has the highest betweenness centrality:

Find the stations with the largest number of neighboring stations in the London Underground network:

Find the messages receiving the largest number of replies in the network of email sent to the MathGroup list in November 2011:

The most interesting subject of the month:

Compute the message generating the largest total number of messages:

## Properties & Relations(11)

GraphHub gives the center of a graph with respect to DegreeCentrality:

With respect to in-degree centrality:

With respect to out-degree centrality:

For simple graphs, GraphHub gives the center with respect to VertexDegree:

Or with respect to VertexInDegree:

Or with respect to VertexOutDegree:

Self-loops are ignored:

For a CompleteGraph, every vertex is a hub:

For a PathGraph, all vertices except for the endpoints are hubs:

For a CycleGraph, every vertex is a hub:

For a WheelGraph of size 5 or more, the hub of the wheel is the graph hub:

For a GridGraph, all vertices that are not at an edge of the grid are hubs:

For a CompleteKaryTree, all vertices except for the leaves and the root are hubs:

The center of a graph with respect to EccentricityCentrality is obtained with GraphCenter:

The set of vertices with maximum VertexEccentricity is obtained with GraphPeriphery:

The BetweennessCentrality center of a graph:

The ClosenessCentrality center:

The EigenvectorCentrality center:

## Possible Issues(1)

Self-loops are not accounted for:

Use VertexDegree to find the center with self-loops included: